Multiplying Fractions

Date: 07/09/99 at 18:00:43
From: Julie-Anne
Subject: How do you multiply fractions?

Hi Dr Math.

I am being home-schooled and I was wondering if you could help me to 
understand how to multiply fractions. I just need to know how to  
multiply fractions like this:  3/5 x 5/8.

Thank you,
Julie

Date: 07/13/99 at 16:59:43
From: Doctor Peterson
Subject: Re: How do you multiply fractions?

Hi, Julie.

There are two important things to know about multiplying fractions: 
how to do it, which is pretty easy, and how to make it easier, which 
takes a little more thought. (Yeah, that sounds crazy, but it's 
generally true that a little more thought at the start makes a lot 
less work in the end.)

The basic idea is simply to multiply the numerators to get the 
numerator of the product, and multiply the denominators to get the 
denominator of the product. For your example,

     3     5    3 * 5   15 <-- product of numerators
    --- * --- = ----- = --
     5     8    5 * 8   40 <-- product of denominators

Now you have to simplify it; the numerator and denominator are both 
multiples of 5, so we can divide them by 5 to get

    15   15 / 5    3
    -- = ------ = ---
    40   40 / 5    8

That's the answer, and you could stop here if you like.

But, if you look closely, you can see that we just wasted a lot of 
work: we multiplied 3 and 8 by 5, and then divided the results by 5 
again. We can save this work by looking at the product before we 
actually multiply, and just cancelling out any common factors, namely 
the 5 we see in both numerator and denominator, knowing we'll just be 
dividing it out in the end anyway:

                    1
                    /
     3     5    3 * 5   3 * 1    3
    --- * --- = ----- = ----- = ---
     5     8    5 * 8   1 * 8    8
                /
                1

So it's good enough to just learn the rule that says multiply and then 
simplify; but if you want to save work, you can think a little more 
before you multiply.

You should look through our archives on fractions to see if any other 
explanations you find will help - sometimes just hearing the same 
thing explained by people with different ways of expressing 
themselves can make it click. Here are the archives:

http://mathforum.org/dr.math/tocs/fraction.middle.html   

and here are some specific questions to start with:

Fraction Basics
http://mathforum.org/dr.math/problems/scamp.7.21.96.html   

Multiplying Mixed Fractions
http://mathforum.org/dr.math/problems/dunphy5.28.98.html   

Multiplying Fractions by Cancellation    
http://mathforum.org/dr.math/problems/washington10.9.98.html   

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/